Three particles,located initially on the vertices of an equilateral triangle of side $L$,start moving with a constant speed $v$ towards each other in a cyclic manner,forming spiral loci that converge at the centroid of the triangle. The length of one such spiral locus will be

If two stones are projected at angles $\theta$ and $(90^{\circ}-\theta)$ respectively with the horizontal with a speed of $20 \ ms^{-1}$. If the second stone rises $10 \ m$ higher than the first stone,then the angle of projection $\theta$ is (acceleration due to gravity $g = 10 \ ms^{-2}$) (in $^{\circ}$)

Two bodies of masses $m$ and $3m$ are rotating in horizontal circles of radii $r$ and $\frac{r}{3}$ respectively. The tangential speed of the body of mass $m$ is $n$ times that of the heavier body. If the centripetal force is the same for both,the value of $n$ is:

$A$ car runs at a constant speed on a circular track of radius $100\,m$,taking $62.8\,s$ for every circular lap. The average velocity and average speed for each circular lap respectively are:

Two particles,one at the centre of a circle of radius $R$,and another at a point $Q$ on the circle,start moving towards a point $P$ on the circle at the same time (see figure below). Both move with uniform velocities $\vec{V}_1$ and $\vec{V}_2$ respectively. They reach the point $P$ at the same time. If the angle between the velocities is $\theta$ and the angle subtended by $P$ and $Q$ at the centre is $\phi$ (as shown in the figure),then:

$A$ merry-go-round starts from rest and accelerates at $0.4 \ rad \ s^{-2}$ for the first $5 \ s$. It then rotates at this constant angular velocity for $30 \ s$ and finally comes to rest with the same magnitude of angular deceleration. What is the total linear distance traveled by a child sitting $3 \ m$ from the center of the merry-go-round?